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An Adaptive Framework for Robust Structural Shape Optimization under Uncertainty

Oğuz Han Altıntaş, Hamdullah Yücel

TL;DR

This work tackles robust shape optimization under uncertainty governed by linear elasticity by formulating the problem on a fixed domain using an ersatz-material approach and seeking to minimize the expected compliance $\mathbb{E}[J(\mathcal{W},\omega)]$. The authors develop an adaptive stochastic optimization framework that jointly controls the MC sample size, spatial mesh, and gradient-step length through a priori and a posteriori error indicators based on the dual-weighted residual (DWR) method, including a domain form of the shape derivative obtained via averaged adjoint techniques. Core contributions include the derivation of the stochastic shape derivative, a multi-goal DWR estimator for both PDE and deformation operators, and an adaptive sampling strategy that updates sample sizes via an augmented inner product test. Numerical experiments on leg-like structures demonstrate substantial computational savings while preserving accuracy, validating the method’s potential for efficient robust design under uncertain loading and material properties. The framework offers a practical path to reliable, efficient design in applications such as aerospace landing gear and legged robotics, where random contact and loading pose significant design challenges.

Abstract

This work proposes an adaptive framework to solve a robust structural shape optimization problem governed by linear elasticity models that account for uncertainties in the loading and material inputs. A posteriori error estimators are constructed to adjust the sample size, mesh size, and step length. The size of the sample set in the stochastic gradient approximation is dynamically determined depending on the variance of the shape derivative. When constructing the a posteriori error estimator in the physical domain, errors arising from the discretization of the deformation bilinear form, which provides a descent direction, are considered, in addition to errors from the discretization of the linear elasticity system. The step length in gradient-based optimization is also adaptively adjusted by estimating the Lipschitz constant of the stochastic shape derivative. Moreover, an analysis of the existence and distributed-form derivation of the stochastic shape derivative is provided. Finally, the proposed estimation-based adaptive stochastic optimization framework is validated on leg-like structural components, demonstrating its effectiveness in minimizing touchdown compliance under uncertain contact forces.

An Adaptive Framework for Robust Structural Shape Optimization under Uncertainty

TL;DR

This work tackles robust shape optimization under uncertainty governed by linear elasticity by formulating the problem on a fixed domain using an ersatz-material approach and seeking to minimize the expected compliance . The authors develop an adaptive stochastic optimization framework that jointly controls the MC sample size, spatial mesh, and gradient-step length through a priori and a posteriori error indicators based on the dual-weighted residual (DWR) method, including a domain form of the shape derivative obtained via averaged adjoint techniques. Core contributions include the derivation of the stochastic shape derivative, a multi-goal DWR estimator for both PDE and deformation operators, and an adaptive sampling strategy that updates sample sizes via an augmented inner product test. Numerical experiments on leg-like structures demonstrate substantial computational savings while preserving accuracy, validating the method’s potential for efficient robust design under uncertain loading and material properties. The framework offers a practical path to reliable, efficient design in applications such as aerospace landing gear and legged robotics, where random contact and loading pose significant design challenges.

Abstract

This work proposes an adaptive framework to solve a robust structural shape optimization problem governed by linear elasticity models that account for uncertainties in the loading and material inputs. A posteriori error estimators are constructed to adjust the sample size, mesh size, and step length. The size of the sample set in the stochastic gradient approximation is dynamically determined depending on the variance of the shape derivative. When constructing the a posteriori error estimator in the physical domain, errors arising from the discretization of the deformation bilinear form, which provides a descent direction, are considered, in addition to errors from the discretization of the linear elasticity system. The step length in gradient-based optimization is also adaptively adjusted by estimating the Lipschitz constant of the stochastic shape derivative. Moreover, an analysis of the existence and distributed-form derivation of the stochastic shape derivative is provided. Finally, the proposed estimation-based adaptive stochastic optimization framework is validated on leg-like structural components, demonstrating its effectiveness in minimizing touchdown compliance under uncertain contact forces.
Paper Structure (17 sections, 4 theorems, 103 equations, 8 figures, 6 tables, 1 algorithm)

This paper contains 17 sections, 4 theorems, 103 equations, 8 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem
  3. PDE constraint: linear elasticity
  4. Objective functional: compliance
  5. Stochastic shape derivative
  6. Numerical approximation techniques
  7. Representation of random fields
  8. Approximation in the parametric space
  9. Descent direction with level-set method
  10. Computation of a practical step length
  11. Approximation in the spatial domain
  12. Adaptive procedure
  13. Numerical experiments
  14. Randomness study
  15. Mesh refinement study
  16. ...and 2 more sections

Key Result

Proposition 1

For a fixed realization $\omega \in \Omega$, the shape derivative of the penalized cost functional $J^P({\mathcal{W}},\omega)$ in eqn:optimization_penalized_fixed is given by where and $u(\cdot,\omega)$ denotes the weak solution of eqn:elastic_PDE_ersatz posed in the spatial domain ${\mathcal{W}}$ for almost every $\omega \in \Omega$.

Figures (8)

  • Figure 1: Initial level-set function defined on the computational domain $\mathcal{D}$, including the prescribed boundary conditions and an applied load force $g$ at $(0,1)$.
  • Figure 2: Final optimized designs obtained for various loading-angle deviations $\kappa_\phi$ using full Monte Carlo sampling on a fixed mesh.
  • Figure 3: Final optimized designs on a fixed mesh using full and adaptive sampling approaches for a loading-angle deviation of $\kappa_\phi = 30^\circ$.
  • Figure 4: Evolution of the mean compliance $\mathbb{E}_{\xi}[J^P]$ and the size of Monte Carlo set $|S_k|$ for deterministic and random loading angles ($\kappa_\phi = 30^\circ$) on a fixed mesh.
  • Figure 5: Evolution of mean compliance $\mathbb{E}_{\xi}[J^P]$ obtained using the fixed coarse mesh, the fixed fine mesh, the adaptively generated mesh with full sampling.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 1: $\varepsilon$-cone property
  • Definition 2
  • Proposition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Proposition 2
  • Remark 3
  • Remark 4